The fifty-six-dimensional module for \(E_ 7\). I: A four form for \(E_ 7\)
type: Item
instance of: scholarly article
title: The fifty-six-dimensional module for \(E_ 7\). I: A four form for \(E_ 7\)
zbMATH Open document ID: 0838.20047
author: B. N. Cooperstein
published in: Journal of Algebra
publication date: 1996-06-02T00:00:00Z
review text: With the aim of classifying maximal subgroups in later work, the author develops a very explicit description of the Chevalley group \(E\) of type \(E_7\) over a field \(K\) of characteristic different from two. He starts with constructing a 4-form \(F\) on the 56 dimensional space \(M= (\bigwedge^2 V) \oplus (\bigwedge^2 V^*)\), where \(V= K^8\) and \(V^*\) is its \(K\)-linear dual. The automorphism group of the form \(F\) is \(E\) and \(F\) generates the ring of invariants \(K[M]^E\). A special collection of 1-spaces in \(M\) is said to consist of singular points. Enriching it with `lines', he gets a Lie incidence geometry whose automorphism group is \(E\) again. Other orbits of 1-spaces is \(M\) are also determined, with their stabilizers.
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20G15
Mathematics Subject Classification ID: 20E28
Mathematics Subject Classification ID: 51B25
Mathematics Subject Classification ID: 20F28
zbMATH DE Number: 763949
zbMATH Keywords: maximal subgroups
zbMATH Keywords: Chevalley groups of type \(E_ 7\)
zbMATH Keywords: 4-forms
zbMATH Keywords: automorphism groups
zbMATH Keywords: ring of invariants
zbMATH Keywords: Lie incidence geometry
zbMATH Keywords: orbits of 1-spaces
reviewed by: Wilberd van der Kallen
MaRDI profile type: Publication
full work available at URL: jabr.1995.1092
OpenAlex ID: W1562049085
DOI: 10.1006/JABR.1995.1092
description: scientific article; zbMATH DE number 763949